The Rays are hopeful he will miss only one start, with manager Kevin Cash telling reporters, "we're looking at it optimistically that maybe it's just some tendinitis we have to manage."
Archer left the game after yielding a one-out double to DJ LeMahieu with the Rays leading 3-0 in the third inning. He allowed three hits and struck out four in 2 1/3 innings.
The Rays said Archer will be evaluated further on Sunday.
Six Tampa Bay pitchers combined to limit the Yankees to five hits, with an injury-riddled bullpen delivering a strong performance in a 4-0 win.
Archer didn't seem too concerned about the injury after the game, saying, "Fortunately it's nothing serious, but there's definitely some tightness in there I needed to get out. ... I'm really glad the bullpen stepped up."
The two-time All-Star, making his first start at Tropicana Field since Tampa Bay traded him to Pittsburgh in July 2018, signed with the Rays as a free agent in February.
Archer missed the entire 2020 pandemic-shortened season after undergoing surgery to relieve symptoms of neurogenic thoracic outlet syndrome.
The Rays recalled Brent Honeywell Jr., who will start Sunday's game against the Yankees.
The Associated Press contributed to this report.